This book is dedicated to our wives Helen and Mary Lou and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Onate and his group at CIMNE for their help, encouragement and support during the preparation process.

Preface

General problems in solid mechanics and non-linearity

1.1  Introduction

1.2  Small deformation solid mechanics problems

1.3  Variational forms for non-linear elasticity

1.4  Weak forms of governing equations

1.5  Concluding remarks

    References

2. Galerkin method of approximation - irreducible and mixed forms

2.1  Introduction

2.2  Finite element approximation - Galerkin method

2.3  Numerical integration - quadrature

2.4  Non-linear transient and steady-state problems

2.5  Boundary conditions: non-linear problems

2.6  Mixed or irreducible forms

2.7  Non-linear quasi-harmonic field problems

2.8  Typical examples of transient non-linear calculations

2.9  Concluding remarks

    References

3. Solution of non-linear algebraic equations

3.1  Introduction

3.2  Iterative techniques

3.3  General remarks - incremental and rate methods

    References

4. Inelastic and non-linear materials

4.1  Introduction

4.2  Viscoelasticity - history dependence of deformation

4.3  Classical time-independent plasticity theory

4.4  Computation of stress increments

4.5  Isotropic plasticity models

4.6  Generalized plasticity

4.7  Some examples of plastic computation

4.8  Basic formulation of creep problems

4.9  Viscoplasticity - a generalization

4.10 Some special problems of brittle materials

4.11 Non-uniqueness and localization in elasto-plastic deformations

4.12 Non-linear quasi-harmonic field problems

4.13 Concluding remarks

    References

5. Geometrically non-linear problems - finite deformation

5.1  Introduction

5.2  Governing equations

5.3  Variational description for finite deformation

5.4  Two-dimensional forms

5.5  A three-field, mixed finite deformation formulation

5.6  A mixed-enhanced finite deformation formulation

5.7  Forces dependent on deformation - pressure loads

5.8  Concluding remarks

    References

6. Material constitution for finite deformation

6.1  Introduction

6.2  Isotropic elasticity

6.3  Isotropic viscoelasticity

6.4  Plasticity models

6.5  Incremental formulations

6.6  Rate constitutive models

6.7  Numerical examples

6.8  Concluding remarks

    References

7. Treatment of constraints - contact and tied interfaces

7.1  Introduction

7.2  Node-node contact: Hertzian contact

7.3  Tied interfaces

7.4  Node-surface contact

7.5  Surface-surface contact

7.6  Numerical examples

7.7  Concluding remarks

    References

8. Pseudo-rigid and rigid-flexible bodies

8.1  Introduction

8.2  Pseudo-rigid motions

8.3  Rigid motions

8.4  Connecting a rigid body to a flexible body

8.5  Multibody coupling by joints

8.6  Numerical examples

    References

9. Discrete element methods

9.1  Introduction

9.2  Early DEM formulations

9.3  Contact detection

9.4  Contact constraints and boundary conditions

9.5  Block deformability

9.6  Time integration for discrete element methods

9.7  Associated discontinuous modelling methodologies

9.8  Unifying aspects of discrete element methods

9.9  Concluding remarks

    References

10. Structural mechanics problems in one dimension - rods

10.1 Introduction

10.2 Governing equations

10.3 Weak (Galerkin) forms for rods

10.4 Finite element solution: Euler-Bernoulli rods

10.5 Finite element solution: Timoshenko rods

10.6 Forms without rotation parameters

10.7 Moment resisting frames

10.8 Concluding remarks

     References

11. Plate bending approximation: thin (Kirchhoff) plates and C1 continuity

requirements

11.1 Introduction

11.2 The plate problem: thick and thin formulations

11.3 Rectangular element with corner nodes (12 degrees of freedom)

11.4 Quadrilateral and parallelogram elements

11.5 Triangular element with corner nodes (9 degrees of freedom)

11.6 Triangular element of the simplest form (6 degrees of freedom)

11.7 The patch test - an analytical requirement

11.8 Numerical examples

11.9 General remarks

11.10 Singular shape functions for the simple triangular element

11.11 An 18 degree-of-freedom triangular element with conforming

     shape functions

11.12 Compatible quadrilateral elements

11.13 Quasi-conforming elements

11.14 Hermitian rectangle shape function

11.15 The 21 and 18 degree-of-freedom triangle

11.16 Mixed formulations - general remarks

11.17 Hybrid plate elements

11.18 Discrete Kirchhoffconstraints

11.19 Rotation-free elements

11.20 Inelastic material behaviour

11.21 Concluding remarks - which elements?

    References

12. 'Thick' Reissner-Mindlin plates - irreducible and mixed formulations

12.1 Introduction

12.2 The irreducible formulation - reduced integration

12.3 Mixed formulation for thick plates

12.4 The patch test for plate bending elements

12.5 Elements with discrete collocation constraints

12.6 Elements with rotational bubble or enhanced modes

12.7 Linked interpolation - an improvement of accuracy

12.8 Discrete 'exact' thin plate limit

12.9 Performance of various 'thick' plate elements - limitations of thin

    plate theory

12.10 Inelastic material bebaviour

12.11 Concluding remarks - adaptive refinement

    References

13. Shells as an assembly of fiat elements

13.1 Introduction

13.2 Stiffness of a plane element in local coordinates

13.3 Transformation to global coordinates and assembly of elements

13.4 Local direction cosines

13.5 'Drilling" rotational stiffness - 6 degree-of-freedom assembly

13.6 Elements with mid-side slope connections only

13.7 Choice of element

13.8 Practical examples

    References

14. Curved rods and axisymmetric shells

14.1 Introduction

14.2 Straight element

14.3 Curved elements

14.4 Independent slope-displacement interpolation with penalty functions

    (thick or thin shell formulations)

    References

15. Shells as a special case of three-dimensional analysis - Reissner-Mindlin

assumptions

15.1 Introduction

15.2 Shell element with displacement and rotation parameters

15.3 Special case of axisymmetric, curved, thick shells

15.4 Special case of thick plates

15.5 Convergence

15.6 Inelastic behaviour

15.7 Some shell examples

15.8 Concluding remarks

    References

16. Semi-analytical finite element processes - use of orthogonal functions

and 'finite strip' methods

16.1 Introduction

16.2 Prismatic bar

16.3 Thin membrane box structures

16.4 Plates and boxes with flexure

16.5 Axisymmetric solids with non-symmetrical load

16.6 Axisymmetric shells with non-symmetrical load

16.7 Concluding remarks

    References

17. Non-linear structural problems - large displacement and instability

17.1 Introduction

17.2 Large displacement theory of beams

17.3 Elastic stability - energy interpretation

17.4 Large displacement theory of thick plates

17.5 Large displacement theory of thin plates

17.6 Solution of large deflection problems

17.7 Shells

17.8 Concluding remarks

     References

18. Multiscale modelling

18.1 Introduction

18.2 Asymptotic analysis

18.3 Statement of the problem and assumptions

18.4 Formalism of the homogenization procedure

18.5 Global solution

18.6 Local approximation of the stress vector

18.7 Finite element analysis applied to the local problem

18.8 The non-linear case and bridging over several scales

18.9 Asymptotic homogenization at three levels: micro, meso

     and macro

 18.10 Recovery of the micro description of the variables of the problem

 18.11 Material characteristics and homogenization results

 18.12 Multilevel procedures which use homogenization as an ingredient

 18.13 General first-order and second-order procedures

 18.14 Discrete-to-continuum linkage

 18.15 Local analysis of a unit cell

 18.t6 Homogenization procedure - definition of successive yield surfaces

18.17 Numerically developed global self-consistent elastic-plastic

     constitutive law

18.18 Global solution and stress-recovery procedure

18.19 Concluding remarks

     References

19. Computer procedures for finite element analysis

19.1 Introduction

19.2 Solution of non-linear problems

19.3 Eigensolutions

19.4 Restart option

19.5 Concluding remarks

     References

Appendix A Isoparametric finite element approximations

Appendix B Invariants of second-order tensors

Author index

Subject index